Given $ m \angle RPS = 4x - 49$, $ m \angle QPR = 8x - 118$, and $ m \angle QPS = 37$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {8x - 118} + {4x - 49} = {37}$ Combine like terms: $ 12x - 167 = 37$ Add $167$ to both sides: $ 12x = 204$ Divide both sides by $12$ to find $x$ $ x = 17$ Substitute $17$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 4({17}) - 49$ Simplify: $ {m\angle RPS = 68 - 49}$ So ${m\angle RPS = 19}$.